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A note on global in-time behavior for the semilinear nonlocal heat exchanger system

Wenhui Chen, Xiaolin Li, Yan Liu

TL;DR

$\,$This work analyzes a semilinear nonlocal heat exchanger system in $\mathbb{R}^n$ with fractional diffusion $(-\Delta)^{\sigma}$ and Fujita-type nonlinearities, establishing global-in-time existence for small data in the super-critical regime $\min\{p,q\}>p_{\mathrm{Fuj}}(n/\sigma)$ and deriving detailed large-time asymptotics in $L^m$ spaces. The authors develop a time-weighted Banach fixed-point framework together with Fourier analysis of the linear coupled system, yielding explicit kernel representations and sharp decay estimates. They show that, for large times, solutions approach diffusion-based profiles governed by the sums of initial data and nonlinearities, with optimal $L^2$-rates and explicit dependence on the coupling parameters $\mu,\nu$. A byproduct of the analysis is sharp lower bounds for the lifespan in sub-critical and critical cases, extending Fujita-type theory to the nonlocal two-environment heat exchanger and connecting to prior work by Treton (2024).

Abstract

We mainly study global in-time asymptotic behavior for the nonlocal reaction-diffusion system with fractional Laplacians which models dispersal of individuals between two exchanging environments for its diffusive components and incorporates the Fujita-type power nonlinearities for its reactive components. We derive a global in-time existence result in the super-critical case, and large time asymptotic profiles of global in-time solutions in the general $L^m$ framework. As a byproduct, the sharp lower bound estimates of lifespan for local in-time solutions in the sub-critical and critical cases are determined. These results extend the existence part of [S. Tréton, SIAM J. Math. Anal. (2024)].

A note on global in-time behavior for the semilinear nonlocal heat exchanger system

TL;DR

This work analyzes a semilinear nonlocal heat exchanger system in with fractional diffusion and Fujita-type nonlinearities, establishing global-in-time existence for small data in the super-critical regime and deriving detailed large-time asymptotics in spaces. The authors develop a time-weighted Banach fixed-point framework together with Fourier analysis of the linear coupled system, yielding explicit kernel representations and sharp decay estimates. They show that, for large times, solutions approach diffusion-based profiles governed by the sums of initial data and nonlinearities, with optimal -rates and explicit dependence on the coupling parameters . A byproduct of the analysis is sharp lower bounds for the lifespan in sub-critical and critical cases, extending Fujita-type theory to the nonlocal two-environment heat exchanger and connecting to prior work by Treton (2024).

Abstract

We mainly study global in-time asymptotic behavior for the nonlocal reaction-diffusion system with fractional Laplacians which models dispersal of individuals between two exchanging environments for its diffusive components and incorporates the Fujita-type power nonlinearities for its reactive components. We derive a global in-time existence result in the super-critical case, and large time asymptotic profiles of global in-time solutions in the general framework. As a byproduct, the sharp lower bound estimates of lifespan for local in-time solutions in the sub-critical and critical cases are determined. These results extend the existence part of [S. Tréton, SIAM J. Math. Anal. (2024)].
Paper Structure (9 sections, 7 theorems, 54 equations)

This paper contains 9 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Linear nonlocal heat exchanger system
  4. Preliminaries in the Fourier space
  5. Sharp asymptotic behavior of solutions for large time
  6. Semilinear nonlocal heat exchanger system
  7. Philosophy of our proofs
  8. Proofs of Theorem \ref{['Thm=Global-Solution']} and Theorem \ref{['Thm=lower-lifespan']}
  9. Proof of Theorem \ref{['Thm=Large-time']}

Key Result

Theorem 2.1

Let $u_0,v_0\in L^{\infty}\cap L^1$ and $\min\{p,q\}>p_{\mathrm{Fuj}}(\frac{n}{\sigma})$. Then, there exists $\varepsilon_0>0$ such that for any $\varepsilon\in(0,\varepsilon_0]$ the semilinear nonlocal heat exchanger system Eq-Semi-Nonlocal-Heat-System with $\sigma>0$ has uniquely determined global for any $1\leqslant m_1,m_2\leqslant+\infty$. Furthermore, they satisfy the equivalent integral equ

Theorems & Definitions (17)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.3
  • Remark 2.6
  • Remark 2.7
  • ...and 7 more